Alternating series have nice properties.
We start with a very specific form of series, where the terms of the summation alternate between being positive and negative.
In essence, the signs of the terms of alternate between positive and negative.
Recall that the terms of the harmonic series come from the harmonic sequence . An important alternating series is the alternating harmonic series:
Geometric series are also alternating series when . For instance, if , the geometric series is
We know that geometric series converge when and have the sum: When as above, we find
A powerful convergence theorem exists for other alternating series that meet a few conditions.
However, we do not ‘‘give-up’’ and immediately conclude that we cannot apply the alternating series test. Rather, consider the long term behavior of . Treating as a continuous function of defined on , we can take its derivative: The derivative is negative for all (actually, for all ), meaning is decreasing on . We can now apply the alternating series test to the series when we start with and conclude that converges; adding the terms with and do not change the convergence.
The important lesson here is that as before, if a series fails to meet the criteria of the alternating series test on only a finite number of terms, we can still apply the test.
Keep in mind that this does not mean we conclude the series diverges; in fact, it does converge. We are just unable to conclude this based on the alternating series test.
While there are many factors involved when studying rates of convergence, the alternating structure of an alternating series gives us a powerful tool when approximating the sum of a convergent series.
- , and
- is between and .
In this case, is called the th remainder of the series.
Here is the basic idea behind this theorem. Say we have an alternating sequence, . Let’s assume the first term is positive, so the second is negative, and so on. We add the first two numbers and get some number . Now is smaller than , because we subtracted something from . Next, we add on the third term, , to get the partial sum . This is bigger than , because the sequence is alternating, but is smaller than , because the sequence is decreasing.
If we know the series converges to some , we can see that we must be bouncing back and forth around as we add and subtract terms. At one point, is larger than , and then subtracting off the next term makes the partial sum smaller than . In other words, the true limit must be between and . Imagine plotting , , and on a number line. (Or, try it yourself with the alternating harmonic series!) can be no further from than whatever the next term in the sequence is. How do we get from to ? By adding (or subtracting) , which takes us back ‘‘across’’ again. In other words, the distance between and can be no more than .
See if you can use these same ideas to prove the alternating series test!
Let’s see an example of approximating an alternating series.
It is an interesting result that the harmonic series, diverges, yet the alternating harmonic series, converges. The notion that simply alternating the signs of the terms in a series can change a series from divergent to convergent leads us to the following definitions.
- A series converges absolutely if converges.
- A series converges conditionally if converges but diverges.
Note, in the definition above, is not necessarily an alternating series; it may just have some negative terms.
Knowing that a series converges absolutely allows us to make two important statements. The first, given in the following theorem, is that absolute convergence is ‘‘stronger’’ than regular convergence. That is, just because converges, we cannot conclude that will converge, but knowing a series converges absolutely tells us that will converge.
One reason this is important is that our convergence tests all require that the underlying sequence of terms be positive. By taking the absolute value of the terms of a series where not all terms are positive, we are often able to apply an appropriate test and determine absolute convergence. This, in turn, determines that the series we are given also converges.
The second statement relates to rearrangements of series. When dealing with a finite set of numbers, the sum of the numbers does not depend on the order in which they are added. (So .) One may be surprised to find out that when dealing with an infinite set of numbers, the same statement does not always hold true: some infinite lists of numbers may be rearranged in different orders to achieve different sums. The theorem states that the terms of an absolutely convergent series can be rearranged in any way without affecting the sum.
This theorem states that rearranging the terms of an absolutely convergent series does not affect its sum. Making such a statement implies that perhaps the sum of a conditionally convergent series can change based on the arrangement of terms. Indeed, it can. The Riemann rearrangement theorem (named after Bernhard Riemann) states that any conditionally convergent series can have its terms rearranged so that the sum is any desired value, including !
As an example, consider the alternating harmonic series once more. We have stated that Consider the rearrangement where every positive term is followed by two negative terms: (Convince yourself that these are exactly the same numbers as appear in the alternating harmonic series, just in a different order.) Now group some terms and simplify:
By rearranging the terms of the series, we have arrived at a different sum!
One could try to argue that the alternating harmonic series does not actually converge to , and here is an example of such an argument. According to the alternating series test, we know that this series converges to some number . If, as our intuition tells us should be true, the rearrangement does not change the sum, then we have just seen that . The only possibility for is then . But the alternating series approximation theorem quickly shows that . The only conclusion is that the rearrangement did, contrary to our intuition, change the sum.
The fact that conditionally convergent series can be rearranged to equal any number is really an incredible result.
While series are worthy of study in and of themselves, our ultimate goal within calculus is the study of power series, which we will consider in the next section. We will use power series to create functions where the output is the result of an infinite summation.